Math Core

A ladder leans against a wall. How high up does it reach? A TV is advertised by its diagonal. How wide is it? Whenever a right angle is involved, the Pythagorean theorem connects the three sides, so you can find any side from the other two. It puts your square root skills to work.

Legs and hypotenuse

In a right triangle, the two sides that form the right angle are the legs. The side across from the right angle is the hypotenuse. It's always the longest side.

A right triangle with legs 6 and 8. The small square marks the right angle; c is the hypotenuse.

The Pythagorean theorem

In a right triangle with legs $a$ and $b$ and hypotenuse $c$ :

a^2 + b^2 = c^2

Why is it true? Picture a square built on each side of the triangle. The theorem says the two smaller squares have exactly the same total area as the big square on the hypotenuse. For a $3$ - $4$ - $5$ triangle: $9 + 16 = 25$ .

Finding the hypotenuse

Worked example: The triangle above

Find $c$ in the triangle above.

\begin{aligned} 6^2 + 8^2 &= c^2 \\ 36 + 64 &= c^2 \\ 100 &= c^2 \\ c &= 10 \end{aligned}

A length can't be negative, so we take only the positive square root.

Finding a leg

If you know the hypotenuse and one leg, substitute and solve for the missing leg. Subtract to get the unknown square by itself.

Worked example: A ladder

A $13$ ft ladder leans against a wall. Its foot is $5$ ft from the wall. How high up the wall does it reach?

The ladder is the hypotenuse (across from the right angle between wall and ground).

\begin{aligned} 5^2 + b^2 &= 13^2 \\ 25 + b^2 &= 169 \\ b^2 &= 144 \\ b &= 12 \end{aligned}

The ladder reaches $12$ ft up the wall.

Common mistake

The hypotenuse goes alone on one side of the equation. A common mistake is to add the hypotenuse's square to a leg's square. Before you start, find the side across from the right angle and call it $c$ .

Answers that aren't whole numbers

Often $c^2$ isn't a perfect square. Then the exact answer is a square root, and you can estimate it with a calculator or between perfect squares.

Worked example: Rounding a square root

A rectangle is $4$ m by $7$ m. How long is its diagonal, to the nearest tenth?

The diagonal splits the rectangle into two right triangles with legs $4$ and $7$ :

c^2 = 4^2 + 7^2 = 16 + 49 = 65, \qquad c = \sqrt{65} \approx 8.1 \text{ m}

Check: $65$ is between $64 = 8^2$ and $81 = 9^2$ , and much closer to $64$ , so a little more than $8$ makes sense.

Is it a right triangle?

The theorem also works backward (this is called the converse). If the side lengths satisfy $a^2 + b^2 = c^2$ , with $c$ the longest side, the triangle is a right triangle. If not, it isn't.

Worked example: Testing side lengths

Is a triangle with sides $8$ , $15$ and $17$ a right triangle? What about $6$ , $9$ and $11$ ?

$8^2 + 15^2 = 64 + 225 = 289 = 17^2$ . Yes, it's a right triangle.
$6^2 + 9^2 = 36 + 81 = 117$ , but $11^2 = 121$ . No.

Tip

Some whole-number sets come up so often they're worth knowing: $3$ - $4$ - $5$ , $5$ - $12$ - $13$ , $8$ - $15$ - $17$ and $7$ - $24$ - $25$ . Multiples work too: $6$ - $8$ - $10$ is double $3$ - $4$ - $5$ .